Solve any quadratic equation. This free quadratic formula calculator finds the roots (real or complex) of ax² + bx + c = 0, plus the discriminant, vertex, axis of symmetry and sum/product of roots.
Enter the coefficients a, b and c. The calculator solves the equation with the quadratic formula and shows real or complex roots, plus the discriminant and vertex. (a can’t be 0.)
Uses the quadratic formula x = (−b ± √(b²−4ac)) / 2a. When the discriminant is negative, the two roots are complex conjugates.
How the Quadratic Formula Calculator Works
Any equation of the form ax² + bx + c = 0 can be solved directly from its three coefficients. This calculator applies the quadratic formula to find the roots — real or complex — along with the discriminant, vertex and other properties of the parabola.
Formula: x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant D = b² − 4ac determines the type of roots: D > 0 gives two distinct real roots, D = 0 gives one repeated real root at x = −b ÷ 2a, and D < 0 gives two complex conjugate roots x = (−b ÷ 2a) ± (√(−D) ÷ 2a)i. The vertex sits at (−b ÷ 2a, f(−b ÷ 2a)), which is also the axis of symmetry x = −b ÷ 2a. By Vieta’s formulas, the sum of the roots is −b ÷ a and the product of the roots is c ÷ a.
- a — coefficient of x², entered in the “a” field (can’t be 0).
- b — coefficient of x, entered in the “b” field.
- c — the constant term, entered in the “c” field.
Example Scenarios
| a | b | c | Roots (x) |
|---|---|---|---|
| 1 | −5 | 6 | 3, 2 |
| 1 | 0 | −4 | 2, −2 |
| 1 | −3 | −4 | 4, −1 |
| 2 | 4 | 2 | −1 (repeated) |
| 3 | −6 | 3 | 1 (repeated) |
| 1 | 2 | 5 | −1 + 2i, −1 − 2i |
Checking Your Work with a Quadratic Formula Calculator
A quadratic formula calculator is quick, but it only solves the equation you give it, so setting up the coefficients correctly matters most.
- Rearrange first. The equation must equal zero, so x² = 3x + 10 becomes x² – 3x – 10 = 0, giving a = 1, b = -3 and c = -10.
- Keep the signs: drop a minus on b or c and a quadratic formula calculator will faithfully solve the wrong equation.
- Enter missing terms as zero. For 2x² – 8 = 0, b is 0.
- If a is zero, the equation is linear and the formula does not apply.
To verify roots, substitute them back: 5 and -2 both make x² – 3x – 10 equal zero. You can also use the sum and product shown by the quadratic formula calculator, since the roots should add to -b/a and multiply to c/a.
The discriminant tells you what to expect. Positive means two real roots, zero means one repeated root, and negative means a pair of complex roots, which the quadratic formula calculator still reports. For whole-number roots, factoring is often faster than a quadratic formula calculator. The Wikipedia article on the quadratic formula shows its derivation.
Quadratic Formula Calculator FAQ
What does the discriminant tell you about the roots?
The discriminant (b² − 4ac) predicts the type of roots before you even solve for them. A positive discriminant means two distinct real roots, zero means one repeated real root, and a negative discriminant means the roots are a pair of complex conjugates.Why can’t a be zero in this calculator?
If a = 0, the x² term disappears and the equation becomes linear (bx + c = 0), not quadratic, so the quadratic formula’s division by 2a would be undefined. Solve a linear equation separately if a is 0.What happens when the discriminant is negative?
The square root of a negative number isn’t a real number, so the two roots become complex conjugates of the form re ± im·i. The calculator still returns both values, with the real part −b/2a and the imaginary part √(−D)/2a.How do you find the vertex of a parabola from a, b and c?
The vertex’s x-coordinate is −b/2a (the same value as the axis of symmetry), and its y-coordinate comes from plugging that x back into ax² + bx + c. The vertex is the parabola’s minimum point if a > 0, or its maximum if a < 0.What’s the relationship between the sum/product of roots and the coefficients?
By Vieta’s formulas, adding the two roots together always gives −b/a, and multiplying them together always gives c/a, regardless of whether the roots are real or complex. It’s a quick way to sanity-check a solution without resolving the full formula.Can this calculator solve equations with repeated (double) roots?
Yes — when the discriminant equals zero, the calculator recognizes the repeated root case and returns a single value x = −b/2a instead of listing the same number twice as if there were two separate roots.Related Calculators
Once you’ve found the roots, the slope calculator can help explore the line segments between key points on the parabola, and the exponent calculator is handy for checking any power terms by hand. For general number crunching alongside your algebra, try the scientific calculator.